Exercise2.6

A matrix \(L \in \Rmm\) is lower triangular if \(l_{ij}=0\) when \(i \lt j\text{.}\) If, in addition, \(l_{ii}=1\text{,}\) then it is unit lower triangular. Show that the product of two unit lower triangular matrices is lower triangular.

Show that if we write the unit lower triangular matrix \(L\) as \(L=I+N\text{,}\) then \(N\) is nilpotent.

Show that

\begin{equation*} L^{-1} = I -N +N^2 -N^3 + \dots. \end{equation*}

Deduce that \(L^{-1}\) is itself a unit lower triangular matrix.

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